saxon math problem solving paper

The Difference That Gets Results

Problem Solving

Saxon Math provides a balanced math curriculum that emphasizes the importance of problem solving in our modern, global, and technological age. Today’s society is facing energy, political, social, environmental, and economic struggles that rival those of any prior generation. The unnamed challenges of tomorrow will likely be similar in magnitude but different in kind. A mathematically literate populace is needed to work past our current struggles and to formulate the solutions for the problems of tomorrow. Will people be prepared by today’s math instruction to handle these challenges? Saxon Math can play a key role in preparing the next generation for the task of solving the difficult problems that lie ahead.

The foundation to problem solving is knowledge. The value of the storehouse of mathematical knowledge and skill in a student’s long-term memory cannot be overemphasized. Without this resource, students will never become adept problem solvers. Saxon Math has always been recognized as a program committed to the fundamentals of math content knowledge. Yet, the key to problem-solving fluency and success is the proper application of that knowledge and skill in the context of particular problems. The National Council of Teachers of Mathematics (NCTM) defines problem solving as “a task for which the solution is not known in advance.”

The Saxon curriculum exposes students to a wide variety of problems, from simple computation-oriented problems, to the open-ended, novel, and multi-step problems that more closely align to the processes employed by mathematicians in the real world.

Saxon Math Course 3 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 47

Name Performance Task 7A

As an apprentice to an architect, you are given the task of designing a floor plan for a 3-bedroom, 2-bath home with a kitchen/dining room, hallway, and living room. Your plan should meet the specified design requirements. Keep in mind that you will be calculating the area of your floor plan. Use the labels given in the design requirements to mark your rooms. A sample design is drawn for you. Use the blank grid for your design, which should be different from the sample.

Design requirements:

• Your floor plan must fit within the grid that is 20 units long and 16 units wide. Each square unit of the grid represents an area of 25 square feet, measuring 5 feet × 5 feet each.

• All rooms must be in the shape of a square, rectangle, triangle, semicircle, or a combination of these shapes.

• The radius of any semicircular area must be a multiple of the length of one square grid unit (5 ft).

• Bedroom 1 (R1) must NOT be larger than 5 units by 5 units in size. • Bedroom 2 (R2) must NOT be larger than 4 units by 4 units in size. • Bedroom 3 (R3) must be at least 3 units by 3 units in size plus 1 semicircular area. • Bathroom 1 (B1) must NOT be larger than 4 square units. • Bathroom 2 (B2) must be 6 square units. • The hall (H) must connect to at least 1 bathroom and 1 bedroom. • The kitchen/dining room (KD) must be twice as long as it is wide. • The living room (L) must have an area of no less than 12 square units plus

1 semicircular area.

Sample floor plan:

L

KD

B2 B1

R3

R2R1

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Lesson

75-2

Understanding p. 5

Analyzing p. 5

Evaluating p. 5

Higher-Order Thinking Skills

See Section Overview 8Math Center Activities 57–61, 62Journal WritingLiterature ConnectionsExtend and Challenge CD:

Activities 1–6

Extensions and Enrichment

capacitycupfull

gallonliterquart

Mathematical Language

Saxon Math 2 1© Harcourt Achieve Inc. and Nancy Larson. All rights reserved.

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L E S S O N

75-2Saxon Math 2

Written Assessment 14 Identifying Gallon, Half-Gallon, Quart, and Liter Containers Estimating and Finding the Capacity of Containers

Lesson Preparation

materials • Teacher Fact Cards (Sets A–H) • Fact Assessment 14-2 (100 Addition Facts) • Written Assessment 14 • 5 school milk containers (4 empty and 1 unopened) • 1-cup liquid measuring cup • Optional: large self-stick tags • 2 one-gallon, 1 half-gallon, and 1 one-quart milk

containers (empty) • one-liter soda pop bottle (empty)

the night before • Collect the various containers needed for this lesson.

Fill the empty school milk containers and a one-gallon container with water.

in the morning • Put the thermometer outside the classroom window for at

least 5 minutes.• Write the following pattern on the Meeting Board:

80, 75, 70, 65, _____, _____, _____, _____, _____, ______

Answer: 80, 75, 70, 65, 60, 55, 50, 45, 40, 35

• Write 83¢ and 3:30 on a card and show it to the Student of the Day. Provide pictures of 10 dimes, 10 nickels, and 10 pennies.

• Write the following in the area labeled “Problem of the Day”:

On Monday, Nancy gave her cat 10 cat treats. Each day she gave the cat one fewer cat treat. How many treats will she give her cat on Wednesday?

Answer: 10 treats – 1 treat – 1 treat = 8 treats

The comprehensive list of TEKS for Lesson 75-2 can be found in Section Overview 8.

Texas Essential Knowledge and Skills

(2.9)(C) select a non-standard unit of measure such as a bathroom cup or a jar to determine the capacity of a given container

Meeting: (2.2)(B), (2.3)(A), (2.3)(C), (2.3)(D), (2.5)(A), (2.5)(C), (2.6)(C), (2.10)(A), (2.10)(B), (2.11)(B), (2.12)(A), (2.12)(C)

Math Center Activity 62: (2.9)(C)

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22. Machinery Two gears are interlocked. One has a radius of 10 centimeters and for each complete rotation, it rotates the second gear 0.77 of a full turn.

a. What is the relationship between the circumferences of the two gears? b. What is the second gear’s radius, to the nearest centimeter?

(23)(23)

15. Optics A diagram of a hyperbolic mirror is shown.The property of a hyperbolic mirror is that if you shine a beam of light from the mirror, the light is reflected toward the focal point. For this mirror, where should you place a view lens to see the converging light? x

y

O

2

4

6 8

2

4

2 4

Asymptote: y = 3_4

x

Asymptote: y 3_4

x

Reflected light

x

y

O

2

4

6 8

2

4

2 4

Asymptote: y = 3_4

x

Asymptote: y 3_4

x

Reflected light

(109)(109)

Students are more apt to learn from the problems they are asked to solve if they can relate to them in their real-life experiences. The National Math Advisory Panel Final Report asserts that “instruction that features the use of ‘real-world’ contexts has a positive impact on certain types of problem solving” (page 49). Accordingly, a special emphasis in the Saxon Math Kits is given to scenarios involving money, time, and food.

As students become older, their experiences expand across many domains. In Saxon’s newest high school editions, special care was given toward creating problems that would interest older students and help them more clearly envision the type of math required within various careers. In addition, cross-curricular problems and problems that incorporate math-to-math connections are prominent in our Saxon textbooks.

15. Cell Phones You pay $10 a month plus $0.30 per minute for your cell phone. You budget $20 each month for your bill. To find the maximum minutes you can use your phone, solve the inequality 10 + 0.3m ≤ 20.

(77)(77)

1

Saxon Course 3 Performance Task 7A

Saxon Algebra 1 Lesson 82 Problem 15

Saxon Geometry Lesson 32 Problem 22

Saxon Algebra 2 Lesson 116 Problem 15

Saxon Math 2 Lesson 75-2 Problem of the Day

Knowledge

2 Saxon. The Difference That Gets Results.

The first post is a collection of problem-solving strategies. NCTM stresses the point that “math curriculums should apply and adapt a variety of appropriate strategies to solve problems.” Each Saxon textbook lesson requires students to employ a strategy from a vast collection of possible strategies in order to solve a novel problem.

The second post of this framework is a consistent 4-step problem-solving process: understand, plan, solve, check. The process promotes the metacognition suggested by NCTM: “Math curriculums should monitor and reflect on the process of mathematical problem solving.” It also promotes the role of estimating in mathematics.

Lesson 35 216B

Problem Solving Discussion

Problem

Montrelyn has a pet dog and a pet cat. Together, the two animals weigh 42 pounds. The dog weighs 14 pounds more than the cat. How much does each pet weigh?

Focus Strategy Guess and Check

Understand Understand the problem.

“What information are we given?”

We are told that the pet dog and cat weigh 42 pounds altogether. The dog weighs 14 pounds more than the cat.

“What are we asked to do?”

We are asked to find the weight of each animal.

Plan Make a plan.

“What problem-solving strategy can we use?”

We can guess and check.

Solve Carry out the plan.

“What is the sum of the two weights we are looking for? What is the difference?”

The sum will be 42 pounds. The difference will be 14 pounds. For a starting point, we can pretend that the pets weigh the same amount.

“If each pet weighed the same amount, what would be their weights?”

Each pet would weigh half of 42 pounds. Half of 42 is the same as half of 40 plus half of 2, which is 20 + 1 = 21 pounds.

“How can we use the ‘halfway’ amount of 21 pounds to help us guess the weights of the pets?”

For the dog, we can add some weight to the amount of 21 pounds. For the cat, we can subtract the same amount from 21 pounds. By increasing/decreasing 21 pounds by the same amount, we keep the sum of the weights at 42 pounds.

“What is a good guess for how much we should increase/decrease the ‘halfway’ amount of 21 pounds?”

If we increase/decrease by 1 pound, the resulting guesses (22 and 20) have a difference of 2 pounds. If we increase/decrease by 2 pounds, the guesses (23 and 19) have a difference of 4 pounds. We notice that the difference is double the amount we increase/decrease. We can try increasing/decreasing by 7 pounds to get a difference of 14 pounds. Our guesses are 21 + 7 = 28 pounds for the dog and 21 – 7 = 14 pounds for the cat.

Check Look back.

“Are our answers reasonable?”

We know that our answers are reasonable because 28 pounds + 14 pounds = 42 pounds, which is the total weight of the pets. The difference is 28 pounds – 14 pounds = 14 pounds, which represents how much heavier the dog is than the cat.

4

Problem-Solving Worksheet

Understand Plan Solve Check

Name

MK(3e)-PSW-130a © Harcourt Achieve Inc. and Nancy Larson. All rights reserved.

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Saxon Math K (for use with Lesson 130-2)

Draw a Picture

Mr. Connolly put 5 markers on the table. Some of the markers rolled off the table onto the floor. James saw that there was only 1 marker left on the table. Show how many markers are on the floor.

How many markers are on the floor?

130A

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Lesson 116 809

Example 5 Application: Population

The population for a certain town for given years is shown in the table.

Year 1991 1992 1995 2000 2006 2007Population 4680 4824 5716 6205 8944 8991

Find the model that best fits the data. Then use the model to estimate the population of the town in 2002.

SOLUTION

1. Understand The x-values are not equally spaced. Use a graphing calculator to help decide which types of regressions to try. For simplicity, the x-values can be the number of years after the year 1990.

2. Plan The data points are somewhat close to forming a line, although a nonlinear model may better fit the data. Compare the R2 values for different types of regression models to find which fit is best.

3. Solve The R2 values are as follows, linear: ≈ 0.9604, quadratic: ≈ 0.9818, cubic: ≈ 0.9868, quartic: 0.9941, exponential: ≈ 0.9764, logarithmic: ≈ 0.8039.

The quartic model is the best fit. The function is y ≈ -0.474x4 + 17.73x3 - 205.718x2 + 1014.48x + 3717.54.

Find the value of the function when x = 12. This can be done by graphing the equation, choosing value from the Calc menu, and typing 12 for X =.

This model estimates the population in 2002 to be about 7081.

4. Check Compare the answer with the numbers in the table to check for reasonableness. 2002 is between 2000 and 2006 and 7081 is between 6205 and 8944, so the answer is reasonable.

Lesson Practice a. Write a quadratic function that models the data shown in the table.

Temperature (°C) -5 -4 -3 -2 -1 0

Number of Campers 8 12 15 19 19 20

b. Write a quartic function that models the data shown in the table.

Week 10 20 30 40 50 60Value of Stock $42 $25 $9 $12 $27 $41

c. Write an exponential function that models the data shown in the table.

Minutes 1 2 3 4 5 6Number of Bacteria 115 168 250 371 547 808

(Ex 1)(Ex 1)a. y ≈ -0.482x2 + 0.018x +19.96a. y ≈ -0.482x2 + 0.018x +19.96

(Ex 2)(Ex 2)

b. y ≈ -0.0000646x4 +0.0088x3 - 0.35x2 + 3.62x+ 32.83

b. y ≈ -0.0000646x4 +0.0088x3 - 0.35x2 + 3.62x+ 32.83

(Ex 3)(Ex 3)c. y ≈ 77.41(1.478)xc. y ≈ 77.41(1.478)x

Math Reasoning

Analyze Would you use the quartic model to predict the population for the year 2012? Explain.

No, the population has been increasing and the quartic model will give a lower population in the future.

Lesson 116 809

Example 5

This example presents a population application which � ts a quartic model.

Extend the ExampleWhich model would be the worst � t for the data? logarithmic

Additional Example 5The population for a certain town for given years is shown in the table.

Year 1998 2000 2001 2004 2006

Pop(mill)

0.75 1.22 1.3 1.5 1.61

Find the model that best � ts the data. Then use the model to estimate the population of the town in 2002.Sample: y ≈ -0.00098x4 + 0.05x3 - 0.96x2 + 8.11x - 24.50; 1.36 million

Lesson Practice

Problems a-dSca� olding Have students � rst identify the common differences or ratios for each data set. Then have students graph the data to ensure that the suggested model is correct.

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The Saxon curriculum provides a framework to aid students in their problem-solving proficiency. The framework rises from the core content knowledge that is so strongly developed within the curriculum. There are four posts from which this proficiency is constructed. These posts form an explicit approach to problem-solving instruction lauded by the National Math Advisory Panel Final Report: “Explicit systematic instruction was found to improve the performance of students with learning disabilities in computation, solving word problems, and solving problems that require the application of mathematics to novel situations” (page 48).

Saxon Math Intermediate 5 Lesson 35 Teacher’s Manual Problem Solving Discussion

Saxon Math K Lesson 130-2 Problem-Solving Worksheet 130A

Saxon Algebra 2 Lesson 116 Example 5

Knowledge

Prob

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Sol

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For the problem illustrated below, the strategy of guess-and-check is suggested. In all, at least ten different strategies—from acting it out, to making an organized list, to working backward—are introduced, practiced, and reinforced.

Prob

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This structure begins in our Math K kit and continues through our Algebra 2 textbook.

Knowledge

3

The third post is the categorization of problem-solving situations into algebraic plots. These plots allow students to see general formats within the problem data so that they can understand the big picture of the situation instead of being overwhelmed by problem details.

The final post is the gradual and purposefully managed progression of mathematical approaches, from the concrete, to the representational, to the abstract (C-R-A). This balanced approach carefully reflects an understanding of cognitive development and the nature of today’s student population.

Example 2

ot raey eno ni %02 esor doohrobhgien a ni esuoh a fo ecirp naidem ehT $288,000. By how many dollars did the median price increase?

Solution

.%001 si tnecrep lanigiro ehT .elbat oitar a ni srebmun nevig eht drocer eW The change is an increase of 20%. We add the increase to find the “new” percent is 120%. We know that the “new” price is $288,000. We are asked for the amount of change in dollars.

20120

c288,000

120c = 20 ∙ 288,000

c20 288,000

120

c = 48,000

Original

Change ( )

New

% Actual count

100

20

120 288,000

c

si ecirp wen ehT 20% greater than the original price and 120% of eht original price. We find that the median price increased $48,000 .

Lesson 35 217

Some word problems compare numbers of objects or sizes of objects.

The containers range in weight from 245 pounds to 160 pounds. The heaviest container weighs how many pounds more than the lightest container?

In comparison problems, one number is larger and another number is smaller. Drawing a sketch can help us understand a comparison story. We will draw two rectangles, one taller than the other. Then we will draw an arrow from the top of the shorter rectangle to extend as high as the taller rectangle. The length of the arrow shows the difference in height between the two rectangles. The two rectangles and the arrow each have a circle for a number. For this problem, the rectangles stand for the weights of the two containers.

A comparison problem may be solved by using a subtraction formula. If we subtract the smaller number from the larger number, we find the difference between the two numbers. Here we show two ways to write a comparison equation:

Larger− Smaller Difference

Larger − Smaller = Difference

In this problem the number missing is the difference, which we find by subtracting.

1

2 14 5 pounds− 1 6 0 pounds

8 5 pounds

We find that the heaviest container weighs 85 pounds more than the lightest container.

Justify Explain how to decide if the answer is reasonable. Sample: Use compatible numbers; since 250 − 150 = 100, the answer is reasonable.

Lesson 35 217








1


8 5 pounds



Lesson 35 217








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8 5 pounds



Lesson 35 217








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8 5 pounds



Lesson 35 217








1


8 5 pounds



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“Write this problem on your paper.”

“Now write the problem using the multiplication symbol.” 5 × 10 = “How did you write ‘10 + 10 + 10 + 10 + 10’ using the multiplication symbol?”

“How can we find out how many pieces of cereal will be in 5 groups of 10?” count by 10’s

“How many pieces of cereal is that?” 50 * Write 5 ¥ 10 = 50 on the board.

* Repeat with 9 groups of 10 = 9 × 10 = 90 12 groups of 10 = 12 × 10 = 120

* Write the following on the board:

3 ¥ 10 = 7 ¥ 10 = 4 ¥ 10 = 1 ¥ 10 = 6 ¥ 10 = 10 ¥ 10 = 8 ¥ 10 = 0 ¥ 10 =

“Who would like to read one of these problems?”

* Have the children read each problem in the following way:“ groups of 10.”

“How many pieces of cereal are there in 3 groups of 10?” 30 * Fill in the answer.

* Repeat with each problem.

“What we did today is called ‘multiplication.’ “

“You have learned how to multiply by 10.”

“We can write these problems another way.”


10¥ 0

10¥ 1

10¥ 2

10¥ 3

10¥ 4

10¥ 5

10¥ 6

10¥ 7

10¥ 8

10¥ 9

10¥ 10

“Which number in each problem do you think tells us the number of groups now?” bottom number

“What is each answer?”

* Write the answers for the problems.

“Do you see a pattern?”

* Allow time for the children to share observations.

“Who would like to share something you learned in math today?”

* Provide 2–3 minutes for sharing. Allow as many children as possible to respond. Provide appropriate feedback and reinforcement.

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n 92

Saxon Math 2 5© Harcourt Achieve Inc. and Nancy Larson. All rights reserved.

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“Put the extra pieces of cereal (macaroni) in the cup.”

Cups

> Distribute a cup of cereal (macaroni) and copy paper to each child. Allow time for the children to count and group the cereal (macaroni).

“How many groups of 10 do you have?”

Copy Paper

* Write an addition problem on the board to show how many pieces of cereal (macaroni) a child has. For example, if the child has 5 groups of 10, write the following on the board: 10 + 10 + 10 + 10 + 10 = “(Child’s name) has groups of 10.”

“Did anyone else have groups of 10?”

“Did anyone have a different number of groups of 10?”

* Write the addition problem on the board.

* Repeat until all possibilities are listed.

“Let’s count by 10’s to find each answer.”

* Write the answer next to each problem.

“These are called ‘equal groups’ problems because we are adding the same number each time.”

“Put your cereal in your cup.”

* Collect the cups.

* Point to one addition number sentence.

“How many groups of 10 do we have?”

* Write the following below the addition number sentence:

groups of 10 =

* Repeat for each of the addition number sentences.

“Mathematicians have a faster way of writing number sentences when the same number is being added.”

“They write equal groups number sentences like this.”

* Write the following below the first addition number sentence:

¥ 10 =

“Writing the × is a short way of writing ‘groups of.’ ”

“Mathematicians call the × the ‘multiplication symbol.’ “

“Let’s write all of the other equal groups problems using the multiplication symbol.”

* Point to another problem.

“How will we write this using the multiplication symbol?”

* Write ¥ 10 = below the problem, filling in the appropriate number of groups.

* Repeat for the other problems.


10 + 10 + 10 + 10 + 10 =

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Saxon Math 2 Lesson 92 New Concept

Saxon Math Course 3 Lesson 67 New Concept Example 2 Saxon Math Intermediate 5 Lesson 35 New Concept

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KnowledgeFor example, the understanding of “equal groups” problems in the primary grades provides the cognitive basis for the ratio box graphic organizers in the middle school textbooks. This unique tool is then the avenue for a deeper understanding of proportionality, a key ingredient for algebraic success.

Read TranslateWrite

a word equation

Draw a Diagram

Write an equation Solve

The Saxon Math programs offer practice using multiple representations and utilizes modeled drawings beginning as early as kindergarten. These increase in complexity as students move through the grade levels. The method is a powerful tool with both routine and non-routine problems.

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Problem Solving Fluency

Name Time

Saxon Math Course 3 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 41

Power-Up Test 23

Math Course 3Facts Find each product.

(x + 3)(x + 3) (x – 3)(x – 3) (x + 3)(x – 3)

(x + 2)(x + 3) (x + 2)(x – 3) (x – 2)(x – 3)

(x + 4)(x + 5) (x – 4)(x + 5) (x – 4)(x – 5)

Problem Solving Answer the question below.

Problem: The figure shown is a closed box for a product packaging project. The box manufacturer needs a design template to create a cut-out that can be folded to make the box. What design can the manufacturer use to construct the product packaging?

UnderstandWhat information am I given?What am I asked to find or do?

PlanHow can I use the information I am given?Which strategy should I try?

SolveDid I follow the plan? Did I show my work?Did I write the answer?

CheckDid I use the correct information?Did I do what was asked?Is my answer reasonable?

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No other curriculum approaches problem-solving practice and assessment in a cumulative fashion. Therefore, Saxon is uniquely able to hold students accountable for their problem-solving ability and to intervene in a meaningful way whenever a problem-solving weakness develops.

Finally, Saxon students enjoy the provision of math reference materials (math offices, math folders, and student reference guides.) These reference materials allow them to proceed with the problem-solving process even when some of the foundational math knowledge needed to solve the problem may not have reached the level of automaticity.

Most schools are fully aware of the effectiveness of Saxon in laying a strong foundation of math content knowledge, but it certainly doesn’t stop there. In the Saxon curriculum, quality problems, a solid problem-solving framework, and a unique approach that supports students in their problem-solving endeavors all work together to provide an effective and balanced approach to problem solving. Implementing Saxon Math in your school or district will help your students be prepared to solve the math-related problems they will face in life and work in this modern, global, and technological age.

Saxon’s instructional approach supports students as they solve the rigorous, noteworthy problems presented in our programs. In addition to stressing math fundamentals and operational fluency, the Saxon curriculum provides even more support for students to become prolific problem solvers. In accordance to NCTM’s recommendation that “problem solving should not be an isolated part of the curriculum but should involve all content standards”, problem solving is not taught in isolation. Saxon’s distributed approach and program structure ensures that students are engaged in meaningful problem solving every day. And every day, the students not only think about math as they solve problems, they also talk about it. Saxon’s systematic approach to problem solving is the type of explicit approach promoted by the National Math Advisory Panel’s Final Report in that it allows “students many opportunities to ask and answer questions and to think aloud about the decisions they make while solving problems” (page 48). Conversations in both Saxon’s primary program, as well as Saxon’s Intermediate and middle school series steer students toward higher levels of thought as they discuss problem scenarios and solutions.

263H Saxon Math Intermediate 4

L E S S O N 4 1

Alternate Strategy Act It Out or Make a Model

Have students use the demonstration clock or their student clocks to model the times that the clock hands will come together.

Problem Solving Discussion

Problem

The hands of a clock are together at 12:00. The hands of a clock are not together at 6:30 because the hour hand is halfway between the 6 and the 7 at 6:30. The hands come together at about 6:33. Name nine more times that the hands of a clock come together.

Focus StrategiesDraw a Picture or Diagram

Find/Extend a Pattern

Understand Understand the problem.

“What information are we given?”

We are told the two hands of a clock are together at 12:00 and at about 6:33.

“Why do the hands of a clock not come together at 6:30?”

At 6:00, the hour hand points to 6, but by the time the minute hand gets to the 6 the hour hand has moved forward halfway to the 7.

“What are we asked to do?”

We are asked to find nine other times that the hands of a clock come together.

“In Lesson 40, what time did we find when the hands came together?”

We found that the hands come together at 1:05.

Plan Make a plan.

“What problem-solving strategies can we use?”

We can draw pictures of clocks to help us find the times that the hands come together. We can look for a pattern in the times found.

Solve Carry out the plan.

“At 2:00, the hour hand points to the 2. About how many minutes later do you expect the minute hand to ‘catch up’ to the hour hand?”

At 2:10, the minute hand will point to the 2, but the hour hand will have moved forward a little bit. We guess that 2:11 is the time when the hands come together. We can draw a picture to help us visualize the problem.

“At 3:00, the hour hand points to the 3. About how many minutes later do you expect the minute hand to ‘catch up’ to the hour hand?”

We guess that the minute hand will catch up by 3:16.

“Can you name other times when the clock hands will come together?”

We can find a pattern in the times we have already named. It is 65 minutes from 12:00 to 1:05, and then 66 minutes to 2:11, and then 65 minutes to 3:16. We might expect that it is either 65 or 66 minutes between times that the hands of a clock come together. The other times are: 4:22, 5:27, 7:38, 8:44, 9:49, and 10:55.

Check Look back.

“Are our answers reasonable?”

We know our answers are reasonable because they follow a pattern and because we know where the hands of a clock point at certain times of the day.

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